First we prove that (a) implies (b). By the definition of a \(Z\) -separating set,
we know that there exists a term ordering \(\sigma \)  on \(\mathbb {T}^n\)  such
that \(z_i =\mathop {\rm LT}\nolimits _\sigma (f_i)\)  for \(i=1,\dots , s\) . This implies \(\mathop {\rm DLog}\nolimits _{z_i}(t)>_\sigma 0\)  for every \(t\in \mathop {\rm Supp}\nolimits (f_i) \setminus \lbrace z_i\rbrace \) , and therefore
\(v >_\sigma 0\)  for every \(v\in \mathop {\rm DLog}\nolimits _Z(F)\) .
Next we note that (b)\(\Rightarrow \) (c) is shown in the theory of Gröbner fans
(see [1]}, Corollary 2.2).
Finally, to prove that (c) implies (a), we observe that the vector \(u\) 
can be taken as the first row of a matrix \(V\in \mathop {\rm Mat}\nolimits _n(\mathbb {Z})\) 
which defines a term ordering \(\sigma \)  on \(\mathbb {T}^n\) .
Then we have \(v >_\sigma 0\)  for all \(v\in \mathop {\rm DLog}\nolimits _Z(F)\) , and therefore
\(z_i >_\sigma t\)  for all \(t\in \mathop {\rm Supp}\nolimits (f_i) \setminus \lbrace z_i\rbrace \)  and \(i=1,\dots ,s\) .
Hence we conclude that \(z_i =\mathop {\rm LT}\nolimits _\sigma (f_i)\)  for \(i=1,\dots ,s\) .
